TMUA Mathematical Reasoning & Logical Thinking 5 — Questions and Answers
Question 1: Which of the following arguments is logically valid?
- P→Q, Q is true, therefore P is true
- P→Q, P is false, therefore Q is false
- P→Q, Q is false, therefore P is false (Correct answer)
- P→Q, P is true, therefore Q is false
Correct answer: P→Q, Q is false, therefore P is false
Modus tollens: if P→Q and ¬Q, then ¬P; this is the only valid argument of the four.
Question 2: A bag contains 4 red and 6 blue balls. Two balls are drawn without replacement. What is the probability both are red?
- 2/15 (Correct answer)
- 4/25
- 1/6
- 2/9
Correct answer: 2/15
P = (4/10) × (3/9) = 12/90 = 2/15.
Question 3: The proposition '∃x ∈ ℤ such that x² = 2' is:
- True, since x = √2 works
- False, since √2 is irrational and not an integer (Correct answer)
- True, since x = 1.41… is close to √2
- False, since no real number squares to 2
Correct answer: False, since √2 is irrational and not an integer
√2 is irrational and not an integer, so no integer x satisfies x²=2, making the statement false.
Question 4: Using the pigeonhole principle, what is the minimum number of students needed to guarantee that at least 3 share the same birth month?
- 13
- 24
- 25 (Correct answer)
- 36
Correct answer: 25
With 12 months, distributing 24 students allows 2 per month; the 25th guarantees a third in some month.
Question 5: Which of the following is the negation of 'All prime numbers greater than 2 are odd'?
- No prime numbers greater than 2 are odd
- Some prime numbers greater than 2 are odd
- All prime numbers greater than 2 are even
- There exists a prime number greater than 2 that is not odd (Correct answer)
Correct answer: There exists a prime number greater than 2 that is not odd
The negation of '∀x, P(x)' is '∃x, ¬P(x)', i.e., there exists a prime greater than 2 that is not odd.
Question 6: If f(x) = 2x + 1 and g(x) = x², what is g(f(3))?
- 13
- 49 (Correct answer)
- 37
- 169
Correct answer: 49
f(3) = 2(3)+1 = 7; g(7) = 7² = 49.
Question 7: A truth table for P XOR Q (exclusive or) shows P XOR Q is true when:
- Both P and Q are true
- P and Q have the same truth value
- Exactly one of P and Q is true (Correct answer)
- Both P and Q are false
Correct answer: Exactly one of P and Q is true
XOR is true precisely when the operands differ in truth value — one true and one false.
TMUA Mathematical Reasoning & Logical Thinking 5
Which of the following arguments is logically valid?